Google searching the two terms show 61% of pages use the term "web site" and the others use "website".

Many pages use both!

Just out of trivial curiosity, does anyone know of an "authority" who argues that it should be one or the other?

Its website one word

A website (or Web site) is a collection of web pages, typically common to a particular domain name or subdomain on the World Wide Web on the Internet

Website - Wikipedia, the free encyclopedia

Even Wikipedia mentions the subject:

As noted above, there are several different spellings for this term. Although "website" is commonly used, the Associated Press Stylebook, Reuters, Microsoft, academia, and dictionaries such as Oxford and Merriam-Webster use the two-word, capitalised spelling "Web site". This is because "Web" is not a general term but a shortened form of "World Wide Web". An alternative version of the two-word spelling is not capitalised. As with many newly created terms, it may take some time before a common spelling is finalised. (This controversy also applies to derivative terms such as "Web master"/"webmaster".)

You say po tay to and I say po tah to. You say to may to and I say to mah to. Po tay to, po tah to, to may to, to mah to oh lets call the whole thing off!

I'm from Ahia. I say Tater :p